Subgroups — C₁₃₇×#10872 #46860258

Order 35072.

730 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#344×C₁₃₇	C₂	1	2	yes	maximal
5	#1733×C₁₃₇	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#108×C₁₃₇	—	2	1	no	—
12	C₂×#108×C₁₃₇	—	2	1	no	—
13	C₂×#108×C₁₃₇	—	2	2	no	—
14	C₂×#108×C₁₃₇	C₄	1	4	yes	complemented
15	#427×C₁₃₇	—	2	2	no	—
16	#427×C₁₃₇	C₄	1	4	yes	complemented
17	C₂×#107×C₁₃₇	—	2	1	no	—
18	C₂×#106×C₁₃₇	—	2	1	no	—
19	#405×C₁₃₇	—	2	1	no	—
20	#405×C₁₃₇	—	2	1	no	—
21	#405×C₁₃₇	C₄	1	2	yes	complemented
22	#405×C₁₃₇	C₄	1	2	yes	complemented
23	C₂×#97×C₁₃₇	—	2	1	no	—
24	C₂×C₄×C₈×C₁₃₇	—	2	1	no	—
25	#344×C₁₃₇	—	2	4	no	—
26	#339×C₁₃₇	—	2	4	no	—
27	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₃₇	—	2	1	no	—
29	C₂×M₁₆×C₁₃₇	—	2	1	no	—
30	C₂×M₁₆×C₁₃₇	—	2	1	no	—
31	C₂×M₁₆×C₁₃₇	—	2	2	no	—
32	C₂×M₁₆×C₁₃₇	—	2	2	no	—
33	C₂×M₁₆×C₁₃₇	—	2	2	no	—
34	C₂×M₁₆×C₁₃₇	—	2	2	no	—
35	C₂×M₁₆×C₁₃₇	—	2	2	no	—
36	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
39	C₂×M₁₆×C₁₃₇	Q₈	1	2	yes	—
40	C₂²×C₈×C₁₃₇	—	2	1	no	—
41	C₂²×C₈×C₁₃₇	—	2	2	no	—
42	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
51	C₂²×C₈×C₁₃₇	Q₈	1	2	yes	—
52	#117×C₁₃₇	—	2	1	no	—
53	#117×C₁₃₇	—	2	1	no	—
54	#117×C₁₃₇	—	2	2	no	—
55	#117×C₁₃₇	—	2	2	no	—
56	#117×C₁₃₇	—	2	2	no	—
57	#117×C₁₃₇	C₂×C₄	1	4	yes	—
58	#117×C₁₃₇	C₂×C₄	1	2	yes	—
59	#117×C₁₃₇	D₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₁₃₇	Q₈	1	2	yes	—
62	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
63	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
64	C₂×C₄²×C₁₃₇	—	2	1	no	—
65	C₂×C₄²×C₁₃₇	—	2	2	no	—
66	#108×C₁₃₇	—	2	4	no	—
67	#108×C₁₃₇	—	2	4	no	—
68	#107×C₁₃₇	—	2	4	no	—
69	#106×C₁₃₇	—	2	4	no	—
70	#97×C₁₃₇	—	2	2	no	—
71	#97×C₁₃₇	—	2	2	no	—
72	C₄×C₈×C₁₃₇	—	2	2	no	—
73	C₄×C₈×C₁₃₇	—	2	2	no	—
74	C₂⁴×C₁₃₇	#41	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₇	—	2	2	no	—
76	C₂²×C₄×C₁₃₇	—	4	2	no	—
77	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
89	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
90	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
91	M₁₆×C₁₃₇	—	2	2	no	—
92	M₁₆×C₁₃₇	—	2	2	no	—
93	M₁₆×C₁₃₇	—	2	2	no	—
94	M₁₆×C₁₃₇	—	2	4	no	—
95	M₁₆×C₁₃₇	—	2	4	no	—
96	M₁₆×C₁₃₇	—	4	2	no	—
97	M₁₆×C₁₃₇	—	4	2	no	—
98	M₁₆×C₁₃₇	C₄⋊C₄	1	4	yes	—
99	C₂×C₈×C₁₃₇	—	2	2	no	—
100	C₂×C₈×C₁₃₇	—	2	2	no	—
101	C₂×C₈×C₁₃₇	—	2	2	no	—
102	C₂×C₈×C₁₃₇	—	2	4	no	—
103	C₂×C₈×C₁₃₇	—	4	2	no	—
104	C₂×C₈×C₁₃₇	—	4	2	no	—
105	C₂×C₈×C₁₃₇	C₄²	1	2	yes	—
106	C₂×C₈×C₁₃₇	C₄²	1	2	yes	—
107	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
112	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
113	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
114	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
115	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
116	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
117	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
118	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
119	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
120	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
121	C₄⋊C₄×C₁₃₇	—	2	2	no	—
122	C₄⋊C₄×C₁₃₇	—	2	4	no	—
123	C₄⋊C₄×C₁₃₇	—	4	2	no	—
124	C₄⋊C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
125	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
126	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
127	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
128	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
129	C₄²×C₁₃₇	—	2	2	no	—
130	C₄²×C₁₃₇	—	2	2	no	—
131	C₄²×C₁₃₇	—	2	4	no	—
132	C₄²×C₁₃₇	—	4	2	no	—
133	C₂³×C₁₃₇	—	4	2	no	—
134	C₂³×C₁₃₇	#117	1	1	yes	characteristic
135	C₂³×C₁₃₇	C₄×D₄	1	1	yes	characteristic
136	C₂³×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
137	C₂³×C₁₃₇	#121	1	1	yes	characteristic
138	C₂³×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂³×C₁₃₇	#124	1	1	yes	characteristic
140	C₂³×C₁₃₇	#125	1	1	yes	characteristic
141	C₂×C₄×C₁₃₇	—	2	4	no	—
142	C₂×C₄×C₁₃₇	—	4	4	no	—
143	C₂×C₄×C₁₃₇	—	4	4	no	—
144	C₂×C₄×C₁₃₇	—	8	4	no	—
145	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
146	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
155	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
162	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
163	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
164	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
165	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
166	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
167	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
168	C₂×C₄×C₁₃₇	#125	1	1	yes	characteristic
169	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
170	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
171	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
172	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
173	C₈×C₁₃₇	—	2	4	no	—
174	C₈×C₁₃₇	—	4	4	no	—
175	C₈×C₁₃₇	#95	1	2	yes	—
176	C₈×C₁₃₇	#95	1	2	yes	—
177	C₈×C₁₃₇	#95	1	2	yes	—
178	C₈×C₁₃₇	#95	1	2	yes	—
179	C₂²×C₁₃₇	—	2	2	no	—
180	C₂²×C₁₃₇	—	2	2	no	—
181	C₂²×C₁₃₇	—	2	2	no	—
182	C₂²×C₁₃₇	—	4	1	no	—
183	C₂²×C₁₃₇	—	4	1	no	—
184	C₂²×C₁₃₇	—	4	1	no	—
185	C₂²×C₁₃₇	—	4	1	no	—
186	C₂²×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
187	C₂²×C₁₃₇	#380	1	1	yes	characteristic
188	C₂²×C₁₃₇	#382	1	1	yes	characteristic
189	C₂²×C₁₃₇	#389	1	1	yes	characteristic
190	C₂²×C₁₃₇	#389	1	1	yes	characteristic
191	C₂²×C₁₃₇	#390	1	1	yes	characteristic
192	C₂²×C₁₃₇	#391	1	1	yes	characteristic
193	C₄×C₁₃₇	—	4	4	no	—
194	C₄×C₁₃₇	—	8	4	no	—
195	C₄×C₁₃₇	C₂×#95	1	1	yes	characteristic
196	C₄×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
197	C₄×C₁₃₇	#380	1	1	yes	characteristic
198	C₄×C₁₃₇	#384	1	1	yes	characteristic
199	C₄×C₁₃₇	#389	1	1	yes	characteristic
200	C₄×C₁₃₇	#389	1	1	yes	characteristic
201	C₄×C₁₃₇	#390	1	1	yes	characteristic
202	C₄×C₁₃₇	#391	1	1	yes	characteristic
203	C₂×C₁₃₇	—	2	1	no	—
204	C₂×C₁₃₇	—	2	1	no	—
205	C₂×C₁₃₇	—	2	1	no	—
206	C₂×C₁₃₇	—	2	1	no	—
207	C₂×C₁₃₇	—	2	1	no	—
208	C₂×C₁₃₇	—	2	1	no	—
209	C₂×C₁₃₇	C₄⋊#95	1	1	yes	characteristic
210	C₂×C₁₃₇	#1814	1	2	yes	—
211	#10872	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₁₃₇	#10872	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²×#108	C₂×C₁₃₇	1	1	yes	characteristic
214	C₂×#427	C₂×C₁₃₇	1	1	yes	characteristic
215	C₂×#405	C₂×C₁₃₇	1	1	yes	characteristic
216	C₂×#344	C₂×C₁₃₇	1	2	yes	—
217	#1733	C₂×C₁₃₇	1	2	yes	—
218	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
219	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
220	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
221	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
222	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
223	C₂×#108	—	2	1	no	—
224	C₂×#108	—	2	1	no	—
225	C₂×#108	—	2	2	no	—
226	C₂×#108	C₄×C₁₃₇	1	4	yes	complemented
227	#427	—	2	2	no	—
228	#427	C₄×C₁₃₇	1	4	yes	complemented
229	C₂×#107	—	2	1	no	—
230	C₂×#106	—	2	1	no	—
231	#405	—	2	1	no	—
232	#405	—	2	1	no	—
233	#405	C₄×C₁₃₇	1	2	yes	complemented
234	#405	C₄×C₁₃₇	1	2	yes	complemented
235	C₂×#97	—	2	1	no	—
236	C₂×C₄×C₈	—	2	1	no	—
237	#344	—	2	4	no	—
238	#339	—	2	4	no	—
239	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
240	C₂×M₁₆	—	2	1	no	—
241	C₂×M₁₆	—	2	1	no	—
242	C₂×M₁₆	—	2	1	no	—
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂×M₁₆	—	2	2	no	—
246	C₂×M₁₆	—	2	2	no	—
247	C₂×M₁₆	—	2	2	no	—
248	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
249	C₂×M₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
250	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
251	C₂×M₁₆	Q₈×C₁₃₇	1	2	yes	—
252	C₂²×C₈	—	2	1	no	—
253	C₂²×C₈	—	2	2	no	—
254	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
255	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
256	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
257	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
258	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
259	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
260	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
261	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
262	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
263	C₂²×C₈	Q₈×C₁₃₇	1	2	yes	—
264	#117	—	2	1	no	—
265	#117	—	2	1	no	—
266	#117	—	2	2	no	—
267	#117	—	2	2	no	—
268	#117	—	2	2	no	—
269	#117	C₂×C₄×C₁₃₇	1	4	yes	—
270	#117	C₂×C₄×C₁₃₇	1	2	yes	—
271	#117	D₄×C₁₃₇	1	2	yes	—
272	C₂×C₄⋊C₄	—	2	2	no	—
273	C₂×C₄⋊C₄	Q₈×C₁₃₇	1	2	yes	—
274	C₂×C₂²⋊C₄	—	2	1	no	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₄²	—	2	1	no	—
277	C₂×C₄²	—	2	2	no	—
278	#108	—	2	4	no	—
279	#108	—	2	4	no	—
280	#107	—	2	4	no	—
281	#106	—	2	4	no	—
282	#97	—	2	2	no	—
283	#97	—	2	2	no	—
284	C₄×C₈	—	2	2	no	—
285	C₄×C₈	—	2	2	no	—
286	C₂⁴	#41×C₁₃₇	1	1	yes	characteristic
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
290	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
291	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
293	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
294	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
297	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
298	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
300	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
301	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
302	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	2	no	—
305	M₁₆	—	2	2	no	—
306	M₁₆	—	2	4	no	—
307	M₁₆	—	2	4	no	—
308	M₁₆	—	4	2	no	—
309	M₁₆	—	4	2	no	—
310	M₁₆	C₄⋊C₄×C₁₃₇	1	4	yes	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	4	no	—
315	C₂×C₈	—	4	2	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	C₄²×C₁₃₇	1	2	yes	—
318	C₂×C₈	C₄²×C₁₃₇	1	2	yes	—
319	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
320	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
321	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
322	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
323	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
324	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
325	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
326	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
327	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
328	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
329	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
330	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
331	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
332	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₄⋊C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₂²⋊C₄	—	2	2	no	—
339	C₂²⋊C₄	—	2	4	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	4	no	—
344	C₄²	—	4	2	no	—
345	C₂³	—	4	2	no	—
346	C₂³	#117×C₁₃₇	1	1	yes	characteristic
347	C₂³	C₄×D₄×C₁₃₇	1	1	yes	characteristic
348	C₂³	C₄×Q₈×C₁₃₇	1	1	yes	characteristic
349	C₂³	#121×C₁₃₇	1	1	yes	characteristic
350	C₂³	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
351	C₂³	#124×C₁₃₇	1	1	yes	characteristic
352	C₂³	#125×C₁₃₇	1	1	yes	characteristic
353	C₂×C₄	—	2	4	no	—
354	C₂×C₄	—	4	4	no	—
355	C₂×C₄	—	4	4	no	—
356	C₂×C₄	—	8	4	no	—
357	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
358	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
359	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
360	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
361	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
362	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
363	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
364	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
365	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
366	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
367	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
368	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
369	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
370	C₂×C₄	C₄×Q₈×C₁₃₇	1	1	yes	characteristic
371	C₂×C₄	C₄×Q₈×C₁₃₇	1	1	yes	characteristic
372	C₂×C₄	C₄×Q₈×C₁₃₇	1	1	yes	characteristic
373	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
374	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
375	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
376	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
377	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
378	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
379	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
380	C₂×C₄	#125×C₁₃₇	1	1	yes	characteristic
381	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
382	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
383	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
384	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
385	C₈	—	2	4	no	—
386	C₈	—	4	4	no	—
387	C₈	#95×C₁₃₇	1	2	yes	—
388	C₈	#95×C₁₃₇	1	2	yes	—
389	C₈	#95×C₁₃₇	1	2	yes	—
390	C₈	#95×C₁₃₇	1	2	yes	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	2	no	—
393	C₂²	—	2	2	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	4	1	no	—
398	C₂²	C₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
399	C₂²	#380×C₁₃₇	1	1	yes	characteristic
400	C₂²	#382×C₁₃₇	1	1	yes	characteristic
401	C₂²	#389×C₁₃₇	1	1	yes	characteristic
402	C₂²	#389×C₁₃₇	1	1	yes	characteristic
403	C₂²	#390×C₁₃₇	1	1	yes	characteristic
404	C₂²	#391×C₁₃₇	1	1	yes	characteristic
405	C₄	—	4	4	no	—
406	C₄	—	8	4	no	—
407	C₄	C₂×#95×C₁₃₇	1	1	yes	characteristic
408	C₄	C₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
409	C₄	#380×C₁₃₇	1	1	yes	characteristic
410	C₄	#384×C₁₃₇	1	1	yes	characteristic
411	C₄	#389×C₁₃₇	1	1	yes	characteristic
412	C₄	#389×C₁₃₇	1	1	yes	characteristic
413	C₄	#390×C₁₃₇	1	1	yes	characteristic
414	C₄	#391×C₁₃₇	1	1	yes	characteristic
415	C₂	—	2	1	no	—
416	C₂	—	2	1	no	—
417	C₂	—	2	1	no	—
418	C₂	—	2	1	no	—
419	C₂	—	2	1	no	—
420	C₂	—	2	1	no	—
421	C₂	C₄⋊#95×C₁₃₇	1	1	yes	characteristic
422	C₂	#1814×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#121
commutator	C₂×C₄	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#380
prime radical	C₂⁴×C₁₃₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
14. C₂×#108×C₁₃₇	C₄	semidirect
16. #427×C₁₃₇	C₄	semidirect
21. #405×C₁₃₇	C₄	semidirect
22. #405×C₁₃₇	C₄	semidirect
211. #10872	C₁₃₇	direct (N × H)
212. C₁₃₇	#10872	direct (N × H)
226. C₂×#108	C₄×C₁₃₇	semidirect
228. #427	C₄×C₁₃₇	semidirect
233. #405	C₄×C₁₃₇	semidirect
234. #405	C₄×C₁₃₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#108×C₁₃₇	2
C₂×#427×C₁₃₇	2
C₂×#405×C₁₃₇	2
C₂×#344×C₁₃₇	2
#1733×C₁₃₇	2
#10872	137

Subgroups — C137×#10872 #46860258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#10872 #46860258